Reconstructible graphs, simplicial flag complexes of homology manifolds and associated right-angled Coxeter groups

TL;DR

This paper explores the relationship between finite graphs, simplicial complexes, and Coxeter groups, establishing that certain graphs derived from homology manifolds are uniquely reconstructible from their subgraphs.

Contribution

It introduces a class of reconstructible finite graphs based on their association with simplicial flag complexes that are homology manifolds.

Findings

Finite graphs as 1-skeletons of homology manifolds are reconstructible.

Established a link between simplicial complexes and graph reconstructibility.

Provided conditions under which graphs are uniquely determined by their subgraphs.

Abstract

In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if $\Gamma$ is a finite graph which is the 1-skeleton of some simplicial flag complex $L$ which is a homology manifold of dimension $n \ge 1$, then the graph $\Gamma$ is reconstructible.